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   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# 线性回归的从零开始实现\n",
    "\n",
    "我们将从零开始实现整个方法，\n",
    "包括数据流水线、模型、损失函数和小批量随机梯度下降优化器"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "origin_pos": 2,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import random\n",
    "import torch\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "根据带有噪声的线性模型构造一个人造数据集。\n",
    "我们使用线性模型参数$\\mathbf{w} = [2, -3.4]^\\top$、$b = 4.2$\n",
    "和噪声项$\\epsilon$生成数据集及其标签：\n",
    "\n",
    "$$\\mathbf{y}= \\mathbf{X} \\mathbf{w} + b + \\mathbf\\epsilon$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "origin_pos": 7,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def synthetic_data(w, b, num_examples):  \n",
    "    \"\"\"生成 y = Xw + b + 噪声。\"\"\"\n",
    "    X = torch.normal(0, 1, (num_examples, len(w)))\n",
    "    y = torch.matmul(X, w) + b\n",
    "    y += torch.normal(0, 0.01, y.shape)\n",
    "    return X, y.reshape((-1, 1))\n",
    "\n",
    "true_w = torch.tensor([2, -3.4])\n",
    "true_b = 4.2\n",
    "features, labels = synthetic_data(true_w, true_b, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "`features`中的每一行都包含一个二维数据样本，\n",
    "`labels`中的每一行都包含一维标签值（一个标量）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "origin_pos": 9,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "features: tensor([-1.4786, -1.1202]) \n",
      "label: tensor([5.0458])\n"
     ]
    }
   ],
   "source": [
    "print('features:', features[0],'\\nlabel:', labels[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "origin_pos": 11,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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    "d2l.set_figsize()\n",
    "d2l.plt.scatter(features[:, (1)].detach().numpy(), labels.detach().numpy(), 1);"
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   "cell_type": "markdown",
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   "source": [
    "定义一个`data_iter`函数，\n",
    "该函数接收批量大小、特征矩阵和标签向量作为输入，生成大小为`batch_size`的小批量"
   ]
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   "execution_count": 7,
   "metadata": {
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    "tab": [
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[-0.2473,  1.5208],\n",
      "        [ 1.0592, -0.6097],\n",
      "        [ 0.5593,  1.6877],\n",
      "        [ 0.4319,  0.8102],\n",
      "        [-0.6606,  1.1467],\n",
      "        [-0.8085,  0.1771],\n",
      "        [ 0.2512, -1.3441],\n",
      "        [ 0.0189, -0.6756],\n",
      "        [ 0.1172,  1.6640],\n",
      "        [ 1.7893,  0.4117]]) \n",
      " tensor([[-1.4637],\n",
      "        [ 8.3978],\n",
      "        [-0.4225],\n",
      "        [ 2.2845],\n",
      "        [-1.0299],\n",
      "        [ 1.9771],\n",
      "        [ 9.2582],\n",
      "        [ 6.5315],\n",
      "        [-1.2173],\n",
      "        [ 6.3933]])\n"
     ]
    }
   ],
   "source": [
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    random.shuffle(indices)\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        batch_indices = torch.tensor(\n",
    "            indices[i: min(i + batch_size, num_examples)])\n",
    "        yield features[batch_indices], labels[batch_indices]\n",
    "\n",
    "batch_size = 10\n",
    "\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X, '\\n', y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "在我们开始用小批量随机梯度下降优化我们的模型参数之前\n",
    "我们需要先有一些参数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "origin_pos": 19,
    "tab": [
     "pytorch"
    ]
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   "outputs": [],
   "source": [
    "w = torch.normal(0, 0.01, size=(2,1), requires_grad=True)\n",
    "b = torch.zeros(1, requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "定义模型，将模型的输入和参数同模型的输出关联起来"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "origin_pos": 22,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def linreg(X, w, b):  \n",
    "    \"\"\"线性回归模型。\"\"\"\n",
    "    return torch.matmul(X, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
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   "source": [
    "定义损失函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "origin_pos": 24,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):  \n",
    "    \"\"\"均方损失。\"\"\"\n",
    "    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "定义优化算法"
   ]
  },
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   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "origin_pos": 27,
    "tab": [
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   "source": [
    "def sgd(params, lr, batch_size):  \n",
    "    \"\"\"小批量随机梯度下降。\"\"\"\n",
    "    with torch.no_grad():\n",
    "        for param in params:\n",
    "            param -= lr * param.grad / batch_size\n",
    "            param.grad.zero_()"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "训练过程"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "origin_pos": 32,
    "tab": [
     "pytorch"
    ]
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.027951\n",
      "epoch 2, loss 0.000110\n",
      "epoch 3, loss 0.000053\n"
     ]
    }
   ],
   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "net = linreg\n",
    "loss = squared_loss\n",
    "\n",
    "for epoch in range(num_epochs):\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X, w, b), y)\n",
    "        l.sum().backward()\n",
    "        sgd([w, b], lr, batch_size)\n",
    "    with torch.no_grad():\n",
    "        train_l = loss(net(features, w, b), labels)\n",
    "        print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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   "source": [
    "比较真实参数和通过训练学到的参数来评估训练的成功程度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "origin_pos": 35,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w的估计误差: tensor([-0.0004,  0.0003], grad_fn=<SubBackward0>)\n",
      "b的估计误差: tensor([0.0007], grad_fn=<RsubBackward1>)\n"
     ]
    }
   ],
   "source": [
    "print(f'w的估计误差: {true_w - w.reshape(true_w.shape)}')\n",
    "print(f'b的估计误差: {true_b - b}')"
   ]
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